Theorem bnj1071 35135

Description: Technical lemma for bnj69 35168. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1071.7	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj1071	⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Proof of Theorem bnj1071

Step	Hyp	Ref	Expression
1		bnj1071.7	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj923 34926	. 2 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
3		nnord 7818	. 2 ⊢ (𝑛 ∈ ω → Ord 𝑛)
4		ordfr 6333	. 2 ⊢ (Ord 𝑛 → E Fr 𝑛)
5	2, 3, 4	3syl 18	1 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3899  ∅c0 4286  {csn 4581   E cep 5524   Fr wfr 5575  Ord word 6317  ωcom 7810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3401  df-v 3443  df-dif 3905  df-ss 3919  df-uni 4865  df-tr 5207  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6321  df-on 6322  df-om 7811

This theorem is referenced by:  bnj1030  35145  bnj1133  35147